Efficiency of Swedish equity funds A DEA approach

Transcription

1 Efficiency of Swedish equity funds A DEA approach Love Westin Love Westin Spring 2017 Master thesis 15 ECTS Economics, Umeå School of Business and Economics, Umeå University, Sweden

2 Efficiency of Swedish equity funds A DEA approach Abstract Author: Love Westin Supervisor: Carl Lönnbark The purpose of this study is to evaluate the efficiency of Swedish equity funds. Funds are evaluated with a model for Data Envelopment Analysis (DEA). The suggested DEA model gives a method to rate equity funds entirely based on economic theory, such as Modern Portfolio Theory and the Efficient Market Hypothesis. Contrary to many other papers, efficiency is here seen from the perspective of a risk averse consumer investing in funds, not a manager managing a fund. Moreover, in this paper the DEA approach is for the first time applied to the Swedish equity market. Previous studies of fund rating methods have found weak or no significant relationship between long term performance of funds and high fund rates e.g. as given by Morningstar rates. The method used by Morningstar is then often criticized for favouring funds with high fees. This paper thus responds to a request in the market for new ways to evaluate fund performance. In the study, a set of frontiers of the most efficient funds in the Swedish market are identified. By comparing the frontiers of efficient funds we have in addition to this identified funds in the Swedish market that both offer low fees and are efficient as well. Key Words: OMX, equity fund, efficiency, Morningstar, fees, management, consumer investment, Data Envelopment Analysis, DEA, linear programming JEL codes: G11, C61, C44

3 Table of Contents 1.Introduction Background Purpose of the paper Delimitations of the study Outline of the paper Review of the literature on fund rating Examples of previous studies Economic theory relevant for the study DEA as a Method of Estimation Introduction to Data envelopment analysis The General Multivariate DEA model The Envelopment DEA program A DEA with return to scale Slack terms The DEA model with VRS and slack variables The data of the study Collection of data Variables in the study Implementation of data into the DEA model Variable properties Test of robustness three models with different inputs The efficiency of the Swedish fund market - results Discussion of the findings Performance of the DEA model The questions raised in the study The validity of the results Suggestions for further research References Appendix A. Efficiency scores and slack variables... 31

4 1. Introduction Background Today, a well-known form of saving money for future consumption is by investing in mutual funds. Investopedia (2017) suggests that individuals with limited knowledge in investment strategies for a reasonable cost should put their investments in funds. This argument is based on the simplicity of investing in funds, compared with the risks involved and knowledge needed for investments in specific assets. To guide investors in their choice of funds, a number of organisations independently rate funds with so called performance adjusted methods. Of those, the most recognisable actor is Morningstar. Rating models such as those used by Morningstar is often based on fund performance data, combined with measures of risk exposure. This study focus on the reliability of those rating methods. In Financial Times, Mooney (2016), criticised these methods, since highly rated funds showed no statistical evidence of being an indicator of future good performance. Moreover, in the article, the Financial Conduct Authority argued that such methods often pushed investors into funds with high manager fees but without any specific extra rewards in return of the investment. Further examples of such criticism, e.g. Strömberg (2015), can be found both with regard to the Swedish market, as well as from the Nobel laureate William F. Sharpe, who in an interview with Dahlberg (2013) about his lecture, The Arithmetic of Active Management, stated that the representative investor should choose index funds over active managed funds with high fees. Because of the large amount of active managed funds in the Swedish market, the Swedish market thus is well suited for this study. Hence, today there seem to be a mismatch between how current rating sites suggest consumers of funds to invest their money, and the recent research about performance of funds. Therefore, this study aims to construct a new fund rating model by considering funds as producers of return of investment, and constructing a best practice frontier with the fairly recent, at least in financial performance evaluation, introduced Data Envelopment Analysis (DEA). Hence, the model in this paper will be set from the perspective of the consumer of funds, based completely on economic theory and thus 1 During the work, many comments and constructive suggestions on the text have been given by my supervisor Carl Lönnbark, teachers at department of economics, and my friends in the course in economics at Umeå University. 1

5 becomes free from other incentives which could interfere with ratings of funds made by an actor, engaged in the market itself. 1.2 Purpose of the paper Given this, the purpose of the paper is to evaluate the efficiency of equity funds on the Swedish market by constructing best practice frontiers with a DEA model. Hence, efficiency is evaluated from a production possibility perspective by defining efficient funds, as those producing a high rate of return on an investment for the least cost. The contribution of the paper is that the analysis is made from the perspective of a risk averse consumer, a consumer that want to invest in efficient funds in the Swedish market. In the study, the cost for a consumer of making an investment in a fund is defined as the combined impact of the management fee and a set of measures of the risk exposures associated with the fund. The paper thus want to answer the following questions: Is it, with the DEA method here suggested, possible to identify differences in efficiency among individual Swedish equity funds? Moreover, is it possible to use the DEA method to identify differences in efficiency among certain groups of managers, e.g. belonging to various banks and institutes? Finally, is it possible to use the DEA method to identify efficient funds that also are low cost funds? 1.3 Delimitations of the study In the study, only funds defined by Morningstar as funds focused on the Swedish market will be considered. The definition of efficiency could have been generalised to include all equity funds, e.g. for different countries and markets, but differences among managers with respect to their incentives may then interfere with the estimated efficiency of the funds. The efficiency of funds would then be based on the underlying specific market reflected by a relevant index, a fact that would introduce a measurement problem, were the performance of a country s stock market rather than the efficiency of managers of funds would have been tested. This sort of causality problem may also be found when mixed industry specialised funds on the Swedish market are studied. In order to reduce the complexity of the problem, this study has been delimitated to broader fund categories excluding certain industry specialised funds. Moreover, because of missing data, it has not been possible to include all funds as categorised by Morningstar during a specific period of time. A selection bias prevails in 2

6 the market of equity funds, this is a result of the quite rapid exit or rebranding of a fund with a performance that is inferior to the market standard. Hence, in order to include funds with a performance relevant for today s market, our focus has been constrained to the years With a longer time period, the data set would have been reduced by funds that have made an exit during the time period. On the other hand, newly started funds would then neither have been included in the data set. A shorter period would have included the new funds but the validity of the result may have been less robust. Observations were also lost by measurement errors or because relevant variables for the study not were included in the Thomson Reuters database Eikon. 1.4 Outline of the paper In chapter 2, a review of relevant literature will be presented. This consists of previous studies and economic theory implemented in the paper. The method of estimation, that is the DEA, will be explained from its basic idea to full implementation in the following chapters. In chapter 4, data and choice of variables will be presented and discussed. Chapter 5 focuses on the implementation of data into the DEA framework. The results from the study are presented in chapter 6. Finally, in chapter 7, the results and the validity of evaluating fund performance with a DEA-best practice frontier are discussed. 2. Review of the literature on fund rating 2.1 Examples of previous studies Today, the ratings made by the company Morningstar is the world leading source for investors to compare performance of funds. Blake and Morey (2000) studied how well the Morningstar funds rating could predict future fund performance. The data in that study covered the years Using regression methods with fixed effects, correlation tests, and a comparison between different indicators of risk, the study found that low rates given by Morningstar often actually also were good indicators of poor future performance of a fund. Instead, among the high rated funds (4-5 stars), the study only found weak statistical evidence for the case that five star funds would outperform four star funds in the future. Moreover, the study found that only at best, a use of Morningstar ratings for the prediction of future performance of funds could outperform other models for prediction of future performance. 3

7 Hence, there is an interest to develop new models that better may predict future performance of funds. In this respect, Malhotra and Malhotra (2013), presents an overview on how to apply DEA methods to evaluate fund efficiency among aggressive growth funds. By the use of cross-sectional data of annual performance from March 2011, 189 US funds are evaluated. An efficiency frontier is constructed with the assumption that managers of aggressive growth funds have the objective to maximise the return of their investments. The cost of production is assumed to be the sum of the cost of management and various risk factors. The main finding of the paper is that 7 of the 189 funds were relative efficient compared with the sample. For a consumer that want to select a fund to invest in, these 7 funds thus are the Pareto efficient choices among all funds in the sample, while the rest are considered to be inefficient choices. Managers of funds may also, as well as consumers of funds, use the previous result as a benchmark for effective management of their own funds. Harslem and Scheraga (2006) evaluated efficiency among Morningstar s small cap funds from a manager perspective. Here the goal of the manager and thus the efficiency of a fund is defined as to maximize the total asset wealth in a fund, this variable is used as the output in the construction of their DEA-model. To find this maximum of assets, managers are assumed to use seven inputs, which include measurements of liquidity, types of asset, fund size and portfolio turnovers. The funds are then divided into three groups based on their efficiency score for closer evaluation of the difference in each input variable. The study analyses a mixture of index and other small cap funds and ultimately finds that only a small proportion of the 58 funds can be considered to be relative efficient given the definition of fund efficiency made in the study. The examples of studies above shows how fund performance with DEA models may be formulated either as a tool for the support of fund managers, or for the consumer s choice of investments among a set of funds. In this study we will, as mentioned, focus on the construction of DEA models that may be a tool for investment decisions made by a risk averse consumer. 2.2 Economic theory relevant for the study Especially two sets of financial theory are of interest for this study, Modern Portfolio Theory (MPT) and the Efficient Market Hypothesis (EMH). The MPT approach was 4

8 initially presented by Markowitz (1952, pp ) in the Journal of Finance. Markowitz paper focused on the optimization of an investment portfolio. The theory is based on the assumption that an investor is risk averse. In an investment case, risk aversion means that the investor tries to increase the return of an investment, its yield, while the risk (the possibility of losses) is kept at a minimum. The trade-off preferences between risk and yield generally varies between individuals. Hence, Markowitz defined a general mathematical formula, based on the mean and the variance in the return of a portfolio with different assets. The goal of the investor is then said to maximize the total portfolio mean return, while minimize the variance of the return. This combined choice can be described Emanuelsson and Marling (2012, pp. 3), as the optimizing problem (1): min (σ 2 Bμ) (1) n Subject to: x i = 1 i=1 x i 0 i = 1,2., n Here, both σ 2 and μ are dependent on x i. The problem in (1) is to minimize the difference between the risk, valued as variance σ 2, and the expected return of investment μ times a scalar B, 0 B. B is an indexation of the risk aversion of an individual. The solution to (1) is a set of x i, which is the share of asset i in the optimal portfolio of assets. The MPT approach is the base for the construction of the efficiency model presented later in this paper. Hence, the assumption of risk averse consumers of funds is also made in the paper. Since the σ 2, and μ are given by the shares of x i, that as such are given by the historic data over funds, our DEA model, in comparison with (1), will identify the choice set B, i.e. the efficiency frontier for different funds of portfolios in the MPT, instead of the x i. Moreover, in our DEA model the rate of return is the output of a fund with historically given shares of assets in x i. This output is in the model assumed to be produced by inputs such as the management cost and various exposures to risk. Criticism made against the MPT often concern, as for example Emanuelsson and Marling (2012, pp. 6), that the standard deviation as a measure of risk gives no information about 5

9 the actual value at risk of an investment. Consumers who makes investment in funds also have to take into account factors such as how much of the invested capital can be lost during a period of time, as well as the market risk premium. To answer questions like what the value at risk of an investment in a specific fund actually is, other variables have been included in the model. Testing the relation between return and a risk free asset, by variables such as e.g. the Sharp ratio, could possibly be implemented in this kind of study, but this aspect has been left for further studies. The other theory mentioned above is the EMH. EMH was presented by Fama (1970, pp ). The theory focuses on the pricing of a specific stock as a function of all relevant information concerning the stock. Initially, Fama described a stock market on a three level efficiency scale (weak-strong). When a market is categorised by weak efficiency, there is no possibilities for a manager of a fund nor a consumer investing in a fund, by use of technical analysis based on historical data, to make predictions of the future price of the stock. This is because the market already have knowledge of this information, and the information is included in the current price. The manager however is still able to make predictions of the stock price by the use of public information like financial reports and insider information which not yet have been taken in account for in the current price. In a market with semi-strong efficiency, this possibility for a manager to use public information for price predictions disappears. However, if insider information is available, the manager could still achieve a higher rate of return than a related market index. On a strongly efficient market finally, a manager of a fund or a consumer choosing between funds simply cannot use any information to create a higher rate of return than a related index. This even includes insider information since such information has become commonly known, and already is included in the current price. If a manager anyhow outperforms an index in a strongly efficient market, this would only be considered as a stroke of luck rather than the result of a successful investment strategy. Even if Fama discusses the efficiency of finance markets from an information rather than a production perspective, the EMH still is relevant for this study. If the market is categorised as strongly efficient, according to the EMH, a manager would only by chance 6

10 produce a better return than the index. This would thus give no incentive for a consumer to invest in fund managers who simply don t mimic the index. In such a case, the outcome of our DEA model would provide us with the index fund with the lowest fee. This alternative should then be chosen by the investing consumer. In such a strongly efficient market the manager would share her objective with the consumer, as noted made by Malhotra and Malhotra (2013). The incentive by the manager would be to increase the total assets of the fund and the consumer would only invest in the cheapest index fund. This would then result in price competition among fund managers, were the price would be defined as the management fee of each index fund. Since, as for example Westin (2015) shows for different factor indexes in Sweden, imperfections on capital markets exists during shorter time periods, a management fee of a fund could thus be seen as a cost of the information seeking made by the manager. The investing consumer could use a higher fee as a reason to expect a higher rate of return on an investment in such a fund. 3. DEA as a Method of Estimation 3.1 Introduction to Data envelopment analysis The DEA approach is based on non-parametric linear programming. The main benefit of applying a DEA model is that only a few specific assumptions have to be made with regard to the data set. However, there is still a need for consideration about the choice of variables in order to construct a proper model. The idea behind DEA is to construct a best practice frontier for different firms or similar actors. In this study those actors are Swedish equity funds. The frontier is constructed by identification of a variable or set of variables that will be defined as the produced output. This is then compared with a set of input variables, representing the cost of production for the output. A DEA thus is comparable with a multivariate method, which makes it possible to use several inputs by weighting their contribution in the production of the output. The weights does not have to be decided a prio in the model, which makes the method easy to implement, Malhotra et al (2016). 3.2 The General Multivariate DEA model A general DEA model consists of a single, or a number of Decision Making Units (DMU), and their input(s), and output(s). In this study, a DMU is a specific fund, inputs are 7

11 variables such as cost of investment and risk exposure, and the output is the return of investment of a fund. A general case with n observations of DMU s, m input (X) and g output (Y) may be described as: Each DMU d, where d = 1. n uses input (X) and output (Y) X i,d for i = 1,2,3. m and d = 1,2,3. n Y o,d for o = 1,2,3. g and d = 1,2,3. n Both inputs and outputs are given weights such as: WX i the weight of input i WY o the weight of output o As explained by Charnes, Cooper and Rhodes (1978, pp. 430), the scalar of efficiency, E1 for the specific DMU 1 is given by the ratio between the output and the required inputs combined with their weights. This results in a fractional programing problem (2), called the CCR model. Subject to: E 1 = max E 1 = g o=1 Y o,d WY o m i=1 X i,d WX i, g o=1 Y o,1 WY o m i=1 X i,1 WX i 1 d = 1,2 n (2) WY, WX > ε The purpose of the first ratio constraint in (2) is to ensure that none of the DMU are given an efficiency larger than 100 percent. The second constraint prevents weights to take a value of zero. In the last constraint, ε, can be given an infinitely small value. By considering (2) with respect to each DMU, we will obtain the relative efficiency between the various units. The obtained vector of efficiency scores E thus only represents the relative efficiency of the n decision making units endogenous to the problem. Hence, efficiency scores may not be compared with other units and organisations exogenous to those involved in a specific formulation of a DEA model. 8

12 If the estimated value of any E equals 1, the DMU is efficient relative to other DMU s in the sample, while if E <1, this DMU is found to be an inefficient actor. Interpretation of the result will thus be based on the specific definition of efficiency given in a specific model. To simplify the calculations, the fractional model formulation in (2) may be transformed into a linear program. Wen (2015, pp. 47) shows this by normalising the input variables and their weights to 1. This results in problem (3) for each DMU. In the case of DMU1 this would give: Subject to: E 1 = Max E g 1 = o=1 WY o Y o,1 (3) m i=1 WX i X i,1 = 1 g o=1 WY o Y o,d i=1 WX i X i,d m 0 d= 1,2 n WY, WX > ε In (3) the input and their respective weights have been normalised to 1 in the first constraint. This linear problem may be compared with the fractional formulation in problem (2). 3.3 The Envelopment DEA program The DEA-model can be set from either an input or output orientation. The choice of orientation should be based on variables and the efficiency definition made in the model (Malhotra and Malhotra, 2013). In our case, MPT governs the choice of variables in the model, where consumers of funds are assumed to be risk averse and tries to minimize the cost of an investment, thus we will here use an input oriented model, the inputs, e.g. cost and risk will be minimized over a fixed output. In order to simplify the algorithm to solve the linear problem in (3), this problem often is replaced by its dual, here called the envelopment form. As is well known, when the primal and the dual both are in their respective optimum, the optimum values will be similar, Bogetoft and Otto (2010, pp. 108). Wen (2015, pp. 47) suggests the dual of the linear 9

13 maximization model (3) to be problem (4). For the linear minimization problem in the case of DMU 1, this would be: 1 = min 1 (4) Subject to: n λ d X i,d 1 X i,1 i = 1,2 m d=1 n λ d Y o,d Y o,1 o = 1,2 g d=1 λ d 0 d = 1,2 n In (4) the model aims to minimize the efficiency of each DMU where λ is a vector of λ d for d = 1, 2. n, corresponding to an element in the vector for each individual DMU. 3.4 A DEA with return to scale The model in (4) is based on the assumption of constant return to scale (CRS). In the previous CCR models, the estimated frontier of the model will have the shape of a straight line. The CRS assumption does not seem reasonable to apply in a financial model, instead the assumption of variable return to scale (VRS) will be applied. VRS simply implies that an increase in input not have to be reflected by the same proportional increase in output as under CRS. The frontier will also take the form of a convex shape. VRS is implemented by adding constraint (5) to the model, as shown by Zhu (2014, pp. 12), based of Banker, Charnes and Cooper (1984). n d=1 λ d = 1 (5) The constraint implies that the sum of the elements in vector λ for each DMU is required to be equal to one. 3.5 Slack terms Two different DMU s can both be placed on the best practice frontier and achieve the efficiency score of 1, even if one firm could relative to another firm decrease (increase) input (output), ceteris paribus. Therefore a so called slack variable may be introduced in 10

14 the model. For an individual DMU, Zhu (2014, pp. 14), defines a slack in inputs respectively outputs by (6) and (7). n s i = X i,1 d=1 λ d X i,d i = 1,2 n (6) n s + o = d=1 λ d Y o,d Y o,1 o = 1,2 n (7) Where s and s + are vectors of input and output slack variables. Positive values in s + would indicate that the specific DMU could increase its output with the indicated value in s +, while the weights (λ) are kept constant. Positive values of s indicates that the input could be decreased with the indicated value in s without changing any of the weights. This implies that if a DMU should be fully efficient it both has to have a given efficiency score equal to one, as well as slackness values equal to zero. 3.6 The DEA model with VRS and slack variables To solve the combined model with VRS and slack variables we will combine the dual of the original linear problem with a separate problem for the calculation of the slackness of each DMU in two steps. Zhu (2015, pp. 16) presents the so called BCC model as problem (8) for DMU 1. 1 = min 1 (8) Subject to: n λ d X i,d 1 X i,1 i = 1,2 m d=1 n λ d Y o,d Y o,1 o = 1,2 g d=1 n λ d = 1 d=1 λ d 0 d = 1,2 n Problem (8) thus is the dual problem (4) combined with the VRS constraint (5). Here in step 1, the efficiency score initially is calculated independent of any slack variables. The second step, will solve the slackness of each DMU. In the case of DMU 1 this would give problem (9): 11

15 Subject to: s m =Max i=1 s i + + s o n s i + λ d X i,d = 1 X i,1 i = 1,2 n d=1 n s λ d Y o,d = Y o,1 i = 1,2 n d=1 n λ d = 1 d=1 g o=1 (9) λ d 0 d = 1,2 n Each DMU will thus be evaluated from its efficiency score and its related slack terms. As stated before, a fully efficient DMU will have an efficiency score of one and slack variables that are zero for all output and input such as s = 0. A DMU with an efficiency score of 1 but with some positive slack terms would instead be considered as weakly efficient. A weakly efficient DMU lies on the efficient best practice frontier, but is considered to be inefficient in the utilization of one or several of the inputs or outputs. 4. The data of the study 4.1 Collection of data Data are collected from the website Morningstar and from the Eikon database (often denoted as the Datastream ) provided by Thomson Reuters. Morningstar is the world largest provider of independent financial research data and investment services. Eikon, is an analyst software for financial data which provides real time fundamental data as for example, of fund performance. The selection of funds were chosen from Morningstar s category Swedish funds. According to Morningstar (2017), this category includes all funds that are actively sold to small time investors in Sweden. The funds chosen have an investment profile towards the Swedish market including some funds that are specialised in small/middle 12

16 sized Swedish firms. Net asset values for funds included in the study have been provided by the Eikon Datastream. 4.2 Variables in the study In the study a set of output and input measures connected with each fund have been collected. The variable Mean rate of return (MRETURN) were chosen as the only output variable in the model, since in a risk free case, the objective of a consumer independent of their attitude towards risk exposure, would be to invest in a fund that maximizes the rate of return on an investment. Results based on estimates of rate of return made on data for only one year, would although be too volatile for any stronger interpretations of manager performance. Hence, an average return of funds has been calculated over 3 years. Three years is therefore the time span of the current study. A longer period would reduce the number of funds in the data set, while a shorter time span, as mentioned would make the results less stable. Four input variables are used. One is a cost variable and three measures risk. The Yearly investment cost represents the investment cost for the consumer. The by Morningstar (2017) calculated average over a longer period is by them given as amounts funding annual cost. It will be used as the measure of a yearly investment cost of a fund. The variable is assumed to be fixed over the three years, and will negatively affect the net return of the investment of the consumer. This measure of costs does however not take into account any hidden costs that can affect the return of an investment, such as courtage and insurance fees. In this study it has not been possible to consider such costs explicitly. Beside the cost of an investment, the investment risk is a central input variable in the MPT. Since the risk of a portfolio is not entirely expressed by e.g. the standard deviation of the return of the portfolio, in this study three variables have been included in order to control for the robustness of various risk measures, in relation to an investment. The first risk variable is the Deviation from Beta of the funds. The β value is the value corresponding to how the rate of return of a specific asset varies with the rate of return of a larger set of related market assets. The β value is calculated by equation (10), French (2016, pp. 2): 13

17 β = Cov variance (r a,r p ) Variance(r p ) (10) Above, β is the ratio obtained from the covariance of the rate of return on the specific asset r a, and the rate of return of the underlying market asset r p, divided by the variance of the rate of return in the underlying market asset. In this study, β has been calculated by the rate of return of the specific fund compared with either the Swedish market indexes, MSCI Sweden or MSCI Sweden small cap for each of the three years. The β values in this study have been calculated by Morningstar. Since a β value of 1 corresponds to a close relation between the specific fund and a related market index, the absolute deviation of β from 1, β 1 is used as an input variable. Deviation from Beta is thus seen as a risk associated with the manager of a fund, were a large deviation from 1, indicate that the manager of a fund to some extent deviates from the market index which the fund has claimed to follow. For the consumer of such a fund, the overall risk of the chosen investment thus will deviate from the expected risk of investment when the fund was bought by the consumer. A second measure of risk is the Standard deviation. The Standard deviation measures the sum of the deviations of the rate of return during a day from the mean rate of return during a period. The Standard deviation is calculated by equation (11), Brown (1982, pp. 938). σ = N i=1 (x i μ) 2 N (11) In (11), σ is the Standard deviation, x i is the single return of day i = 1, 2... N while μ is the mean value of all daily returns during the three years. Since the studied time period range from 2014 until 2016 and includes data on all open trading days, the population variation has been used in the definition. The Standard deviation indicate the pure risk of volatility, since large fluctuations increase the possible return but also the possibility to obtain a loss. The measure contains the total volatility risk, both the market and nonmarket volatility risks. Generally, in a comparison between two funds with the same return of investment, the fund with a lower Standard deviation should be the preferred choice for investment. 14

18 The third measure of risk as an input variable is the Expected shortfall, calculated from the data over the three years. The variable is a measure of the risk associated with the expected daily worst case loss. The Expected shortfall risk, measured by the distribution of the daily return of a fund during the three years. The average of the five percent worst daily returns of a fund gives the Expected shortfall of this fund. 4.3 Descriptive statistics As has been mentioned, the time period of the study is the years This is not a perfect period of time from a volatility perspective. Generally, as seen in the graph below, the period may be considered as a bull market for the Swedish capital market. It does not include extended periods of negative development. Figure 1. The development of OMXSPI over the time period Source: NASDAQ. Figure 1 illustrates how the Swedish stock market all share index (OMXSPI) has had a positive trend during the period. The highest return of a fund that followed the index since the beginning of 2014, was 30 %. This was achieved for an investment made at the start of the period and sold , while the lowest return was achieved for a fund bought at the start and sold , with a decrease by 3 %. The funds have been categorised in three sample groups. Group 1 consists of funds categorised as Swedish funds, dominated by large companies. Group 2 consists of the Swedish small/mid cap funds and Group 3 consists of both funds in Group 1 and 2 and thus the total Swedish market. 15

19 Table 1. Descriptive statistics over mean values of the variables sorted by Group. Variable Group 1 n=80 Group 2 n=23 Group 3 n=103 MRETURN 11,87 19,33 13,56 Yearly investment cost % 1,10 1,57 1,17 Deviation from Beta 0,04 0,08 0,05 Standard deviation % 10,3 16,1 11,7 Expected shortfall % 2,41 2,31 2,38 A comparison between Group 1 and 2 in Table 1, shows that Group 1 has had an average lower MRETURN over the years, but also on average, are less expensive for an investor. Considering the manager risk exposure, funds in Group 1 have a stronger correlation with their related index than funds in Group 2, as seen by the indicated values of Deviation from Beta, that are closer to zero for Group 1. The Standard deviation also indicates that the returns for funds in Group 1 have been less volatile, which would indicate a lower risk. In opposite to this, the Expected shortfall indicates that the daily worst case loss, instead is larger in Group 1. As it should, the combined Group 3, i.e. the Swedish fund market as whole, has values for return and risk in between the first two groups, and shows an average of 13,56 % in MRETURN over the studied years. The average Yearly investment cost of the funds are 1,17 % of the amount invested. The funds in Group 3 are closely related to their market indexes, as expected, with a 0,05 mean in Deviation from Beta. The Expected shortfall of the combined group, is approximate 2,3 %. Group 3 also shows an average Standard deviation of 11,7 %, which measures the volatility of the daily returns during the period. Instead of presenting our individual data for each Group, we will here only illustrated data for Group 3, where all funds from both Group 1 and 2 are included. 16

20 Figure 2. Scatterplot of MRETURN and Yearly investment cost for Group 3. Figure 3. Scatterplot of MRETURN and the Yearly investment cost adjusted with the Expected shortfall for Group 3. Figure 4. Linear fit of MRETURN and Standard deviation for Group 3. Figure 5. Scatterplot of MRETURN and Deviation from Beta for Group 3. As seen by Figure 2, there is no clear relationship between the MRETURN and the Yearly investment cost given in percentage. The relation could although be generalized to a nonlinear exponential fit, as well as, a linear fit although with weak correlation. Still the large variation in the sample prevents the possibility to establish a significant pattern. The figure illustrate that most sample funds take a management cost of 1,17 % of total investment each year. However, the variation in MRETURN increases rapidly among funds with a management cost higher than the sample mean. In Figure 3, the relationship between the MRETURN and the funds Yearly investment cost adjusted by Expected shortfall is shown. In comparison with Figure 2, the Expected 17

21 shortfall of a fund is added as a possible cost of investment, combined with the Yearly investment cost. The relationship could be generalised as a possible non-linear exponential fit, but indicates large influential outliers. As seen in comparison with Figure 2, the observations are more centred in a cluster, but with some funds largely deviating from the cluster. This could be caused by funds that charge high fees who still overlooks the risk imposed by an Expected shortfall, or by low fee funds where the fee also reflects the possibility to get a large shortfall in the investment. Figure 4 describes the risk measured as the Standard deviation against MRETURN. The fitted line identifies a clear positive relation between the variables. An increase in the variation of the mean daily return thus generally results in a higher MRETURN. Finally, Figure 5 shows the Deviation from Beta against MRETURN. Since the values of β 1 are rounded to one decimal, observations are located in fixed clusters. More precise values were not considered as indicators of extended risk exposure. Large values in the Deviation from Beta, around 0,2, indicate a large deviation in return compared with the specific underlying market index. Those thus contain a high manager risk and ideally such funds should have been sorted out and included in other fund categories. This would then result in even fewer sample observations with large values of Deviation from Beta. Considering the sample observations in Figure 5, funds with a Deviation from Beta of 0 are centred around the MRETURN mean value of 12,24, with a positive skewness. The second cluster of observations with the value 0,1 has in comparison to the rest of the sample a large variation in MRETURN and a slightly higher mean of 14,9. The last and above discussed cluster of observations with a Deviation from Beta around 0,2 has a mean MRETURN of 14, but considering the small set of observations, no robust conclusions can be drawn. 18

22 Table 2. Correlation matrix between the variables in the study, based on Group 3. MRETURN Yearly investment cost Standard Deviation Expected Shortfall Deviation from Beta MRETURN 1 Yearly investment cost Standard Deviation Expected Shortfall Deviation from Beta 0,46 1 0,95 0,44 1-0,01-0,09 0,01 1 0,24 0,40 0,22-0,15 1 In Table 2, the correlations between all variables in the study are displayed. It may be seen that MRETURN has a weak positive correlation with the Yearly investment cost but an almost perfect correlation with the Standard deviation. Yearly investment cost and the Standard deviation share a weak positive correlation. Expected shortfall has an almost non existing correlation with all other variables, although the highest correlation is found with respect to the Deviation from Beta. The Deviation from Beta is in turn most correlated with the Yearly investment cost. This correlation is probably caused by actively managed funds, which often demand higher fees for their individual investment strategies. 5. Implementation of data into the DEA model 5.1 Variable properties In (8) (9), the DEA model was specified. As discussed, the measure of efficiency in a DEA model is model specific, and given by the variables included. To be able to draw conclusions from the results, some important aspect about the model thus have to be clarified. A firm in a DEA model is considered as efficient, when relative to other organisations in a sample, no inefficiency is found by its utilization of the defined input/output. By comparing the position of a firm in the set of input/output combinations relative to the efficiency frontier, the efficiency of the firm may be identified (Harlshlem and Scheraga, 2006, pp. 87). The result from a DEA thus is heavily dependent on the choice of variables and the type of firms considered. Hence, we will below explain how the general DEA assumptions are implemented on the funds and the variables chosen in this paper. 19

23 First of all, the homogeneity of the funds included in a model has to be evaluated. According to Bowlin (1998 pp. 19), homogeneity simply means that the funds have to share similarities in their objectives and line of work. Above we introduced β 1 as a measure of manager risk, indicating how well a fund follows its relevant index. Hence, it measures the homogeneity of the funds in a sample. A fund with a large deviation may be managed by other incentives than comparable funds. Consumers making investments can be diverse both in their investment strategies and risk aversion but at the end of the day, the objective of each investment can be simplified to maximizing the yield/rate of return on investments while keeping an as low risk exposure as possible. Thus, this homogeneity among the consumers makes the study possible within a DEA approach. Secondly, the model should reflect isotonic properties. Bowlin (1998, pp. 17) argues that an increase in the input should result in at least a logical positive effect on the output. From the scatter plots in the Figures 2-4 and the correlation table in Table 2, we observe that there at least could exist a logical positive relationship between the inputs and the output for the variables in the model since the variables Standard deviation and Yearly investment cost both indicate positive effects on the MRETURN. In Figure 3, by adding the Expected shortfall to the Yearly investment cost, the relationship with the output, MRETURN, reveals a, possible non-linear, but still positive effect on the output. The relation between the variable Deviation from Beta and the output variable MRETURN in Figure 5, although shows no simple visual isotonic property for this sample. The variation between the different clusters of observations within groups in this sample is so large so that a single significant positive relation is difficult to identify. However, the Deviation from Beta is a measurement of managerial risk and higher values indicate that the output has been produced only through investments deviating from the claimed index in order to obtain a higher MRETURN. The possibility to make such investments, which deviate positively from the market index, is a sign of an isotonic property in the relation between the variables. However, this generally only is possible in the short run, and not in an efficient market, as is indicated by the large variation in the observations in Figure 5. Finally, the combined DEA-model suggested by (8) and (9) are not commendable with non-positive values. To solve this problem, Bowlin (1998, pp. 17) suggests to increase all variables with possible negative values with a fixed constant for, in this case, all funds. The constant must be large enough to eradicate all chances of either zero or negative 20

24 values in the variable. Since the constant is added to each fund, the convexity of the set remains unchanged and produces the same frontier. Hence, a constant value of 1 has been added to the two input variables Yearly investment cost and Deviation from Beta in order to prevent an input of zero. 5.2 Test of robustness three models with different inputs DEA models may be sensitive with respect to the choice of variables and outliers. Since all variables are measured by independent factors, even considering the risk variables, one single model could have been composed. However, instead three alternative formulations have been made in order to test for robustness of the results. First, the variable Deviation from Beta is measured from different indexes for Group 1 and 2. In the combined Group 3, the whole Swedish market, the calculations of β would be misguiding as an input variable. This is because the small/mid cap funds, with low market influence, would be given an overestimated risk measure, and therefore the variable has been removed from Group 3. Moreover, Bowlin (1998, pp. 19) refers to the control of the weighs given to each variable. Here, a combined model with three out of four input variables measuring risk, could put too much weight on the risk aspect of a fund, compared with the single measure of the cost of investment. In order to be able to analyse this, more than one DEA model for each group of funds have been constructed, each with different measurements of risk. Altogether, four different models have been composed. In Model 1 the output MRETURN is produced by three input variables; Yearly investment cost, Standard deviation and Deviation from Beta. This model is used for Group 1 and 2. While Model 2 consist of the same base, but the risk variable Standard deviation has been replaced by the short term risk variable, Expected shortfall. Model 2 is also used with Group 1 and 2. In Model 3 MRETURN is used only with the Standard deviation and the cost variable while in Model 4 the Standard deviation is replaced by the Expected shortfall. Both those models are used for the larger Group 3. In neither of the two Models for Group 3 the Deviation from Beta is included as a measure of risk, as discussed above. This means that we have used six different DEA formulations in this test for efficiency in the Swedish fund market. 21

25 6. The efficiency of the Swedish fund market - results The result from the test of efficiency with the six models are very interesting. Among the sample with 80 funds categorised in Group 1, the broader Swedish funds, it was found that with Model 1, 11 funds are fully efficient. For the same group, Model 2 instead found 13 fully efficient funds that were spanning the best practice frontier. In Table 3 those funds are listed. In Group 2, with 23 Swedish small/mid cap funds, the study found that 8 funds were fully efficient with Model 1. For the same group, Model 2 found 6 funds to be fully efficient. Hence, in this smaller and perhaps more homogeneous sample a larger share, around 30 percent of the funds, were fully efficient. This is a larger share compared with that in Group 1. For Group 3, the Swedish fund market as whole, with 103 funds, 10 respective 8 fully efficient funds were found by Model 3 and Model 4. In Model 3, 3 of the 10 fully efficient funds belong to the small/mid cap category in Group 2 and in Model 4, 4 of the 8 fully efficient funds belong to this small/mid cap category. In Table 3, all funds that span the best practice frontier for each of the four model formulations are represented by a column for the three groups of samples. Those fully efficient funds are Pareto efficient relative to other funds within each Group and model formulation. The remaining funds are thus found to be inefficient. The complete presentation of the efficiency scores and slack terms of each fund may be found in Appendix A. Among the efficient funds in Group 1 and DEA Model 1, Norron active R strictly has the highest MRETURN while Danske Invest Sverige Utd instead has the strictly lowest Standard deviation. Those two funds are thus what may be considered as two corner solutions to the model. In comparison with Model 1, Model 2 has a large portion of index funds among the funds in the best practice frontier. In Model 2, Spiltan aktiefond stabil is considered as a corner solution. This fund has the strictly lowest value among all funds in the sample with respect to the variable Expected shortfall. As can be seen in the table, when comparing this with the same sample in Model 2, the new frontier consists of 4 22

26 funds that also may be found in Model 1. We will return to how this should be interpreted below. Table 3. Fully efficient funds on the Swedish market in each group and model. G1 Model 1 n=80 *Norron active R Spiltan aktiefond investmentbolag Aktie-Ansvar Sverige A Lannebo Utdelningsfond Swedbank Robur Humanfond G1 Model 2 n=80 *Norron active R Spiltan aktiefond investment-bolag Lannebo Sverige Plus Carnegie Sverigefond *Spiltan Aktiefond Stabil Open Fund G2 Model 1 n=23 Carnegie Smabolags fond *Humle Smabolagsfond *AMF Pensions Aktiefond - Smabolag Catella Smabolagsfond Inside Sweden G2 Model 2 n=23 Carnegie Smabolags fond *Humle Smabolags-fond *AMF Pensions Aktiefond - Smabolag Catella Smabolagsfond *ODIN Sverige C G3 Model 3 n=103 Carnegie Smabolags fond *Humle Smabolagsfond AMF Pensions Aktiefond - Smabolag Catella Smabolagsfond Spiltan aktiefond investmentbolag G3 Model 4 n=103 Carnegie Smabolags fond *Humle Smabolags-fond AMF Pensions Aktiefond - Smabolag SEB Sverige Smabolag Chans/Risk-fond Open Fund Spiltan aktiefond investment-bolag Nordnet Superfonden Sverige Handelsbanken Sverige Selektiv (A1) *Danske Invest Sverige Fokus SEB Sverige Smabolag Chans/Riskfond Lannebo Utdelningsfon d *Spiltan Aktiefond Stabil Nordea Indexfond Sverige utd XACT OMXSB Utdelande Swedbank Robur Ethica Sverige Mega Ethos Aktiefond *Danske Invest Sverige utd - - Folksam LO Vastfonden Open Fund Handelsbanken Sverige Index Criteria Handelsbanken Sverigefond Index SPP Aktiefond Sverige A Nordnet Superfonden Sverige Lansforsakringar Sverige Indexnara Swedbank Robur Humanfond Ohman Smabolagsfond B Evil Swedish Small Cap A Ethos Aktiefond Nordnet Superfonden Sverige XACT OMXSB Utdelande Swedbank Robur Humanfond *Danske Invest Sverige utd Nordnet Superfonden Sverige Swedbank Robur Humanfond SPP Aktiefond Sverige A Lansforsakringar Sverige Indexnara Stars * indicates funds with strictly highest/lowest value of either output/input in the group sample. - In both models of Group 2 with small/mid cap funds, Humle smabolagsfond strictly has the highest MRETURN in the sample and AMF Pensions aktiefond smabolag strictly has the lowest Yearly investment cost. In Group 2 and Model 1 Danske invest Sverige focus has the strictly lowest Standard deviation, while in Model 2, Odin Sverige C also is a corner solution with the strictly lowest value of the variable Expected shortfall. In Group 23